Stability on 3D Boussinesq system with mixed partial dissipation - Lin Hongxia; Liu Sen; Guo Xiaochuan; You Ruiqi (Advances in Nonlinear Analysis)
In the article, we are concerned with the three-dimensional anisotropic Boussinesq equations with the velocity dissipation in x2{x}_{2} and x3{x}_{3} directions and the thermal diffusion in only x3{x}_{3} direction. When the spatial domain is the whole space R3{{\mathbb{R}}}^{3}, the global well-posedness and stability problem for the partially dissipated Boussinesq system remain the extremely challenging open problems. Attention here focuses on the periodic domain Ω=R×T2Ω ={\mathbb{R}}× {{\mathbb{T}}}^{2}. We aim at establishing the stability for the problem of perturbations near hydrostatic equilibrium and the large-time behavior of the perturbed solution. We first obtain the global existence of some symmetric fluids in H2(Ω)H^{2}\left(Ω ) for small initial data. Then the exponential decay rates for the oscillations u˜\widetilde{u} and θθ in H1(Ω)H^{1}\left(Ω ) and the homogeneous Sobolev space Hv2˙(Ω)\dot{H_{v}^{2}}\left(Ω ) are also shown. The proof is based on a key observation that we can decompose the velocity uu into the average u¯\overline{u} on T2{{\mathbb{T}}}^{2} and the corresponding oscillation u˜\widetilde{u}. This enables us to establish the strong Poincaré-type inequalities on u˜\widetilde{u}, u3,θ{u}_{3},θ and some anisotropic inequalities, which ensure the establishment of the closed priori estimates. In addition, we also prove the oscillations in one direction u˜(2),u˜(3){\widetilde{u}}^{\left(2)},{\widetilde{u}}^{\left(3)} in H1(Ω)H^{1}\left(Ω ) decay to zero exponentially.